Lie algebra from Dynkin diagram

Here is the construction I gave in class for a Lie algebra given its Dynkin diagram.

Next will be some stuff about Weyl groups and Weyl chambers, delaying the fact that simple reflections generate N(T)/T.

We're going to do other chapters from here next, though in a different order: first the Bruhat decomposition, then stuff about Weyl chambers, filling the hole mentioned above, and then the Borel-Weil theorem that lets us construct all the irreps for complex reductive Lie groups.

April 14

If K is compact with discrete center, pi_1(K) is finite. Hence there are finitely
many Lie groups with that Lie algebra. Proof: use Hurewicz to connect to H^1, and Lie algebra cohomology to show that H^1 vanishes over R.

The angle between any two roots must be 90, 120, 135, 150, or 180.

There are at most two root lengths in a connected root system, and if two they differ by a factor of sqrt(2) or sqrt(3).

Next time: simple systems, Dynkin diagrams, their classification.

Old notes

Here are my notes from a previous incarnation of this class. The bottom one is the one we followed to get to root systems.

Feb 15, 17, 21

Gel'fand-Cetlin patterns.
Theorem: they compute weight multiplicities iff V_lambda (for GL(n)) decomposes as the sum of V_mu (for GL(n-1)) where mu intersperses between lambda.
(We'll prove this later.)

Weyl integration formula.

Bruhat decomposition for GL(n).

The character of T acting on a polynomial ring is formally a rational function.

The weight multiplicities in V_lambda are bounded above by those in a certain polynomial ring.

That's not S_n invariant; so let's hit it with S_n to symmetrize it.
Theorem next time: the Weyl character formula, which says that that's the right answer.

Feb 10

Thm. Multiplicity diagrams of U(n) reps are S_n-symmetric.
Proof. Use S_n to define maps between the weight spaces.

Thm.
1. If V is pointed at lambda, then V contains a unique irrep pointed at lambda.
2. If V,W are pointed at lambda,mu, then V tensor W is pointed at lambda+mu.
3. Alt^k(C^n) is pointed at (1,...1,0,...0) with k 1s.

Thm.
For every weakly decreasing lambda, there exists a unique irrep pointed at lambda, and these are all the irreps of U(n) or GL(n).

Proof. Tensor together Alt^ks to produce a rep pointed at lambda.
Let V be any rep; we'll try to write it as a sum of these V_lambda.
It's enough to check U(n)-isomorphism, and for that, it's enough to check the character, i.e. the multiplicity diagram.
Prove inductively that we can write any S_n-symmetric Z-valued function on Z^n, of finite support, as a linear combination of characters of these V_lambda.
Since irreducible characters are orthogonal, there can't be any other irreps.

Feb 3,8

Representations of tori. Weights. Weight multiplicity diagrams.
The Lie algebras of GL(n), SL(n), U(n), SU(n).
Weyl's unitary trick for those groups. Corollary: complete reducibility.
Thm: the irreps of SL(2) are exactly the Sym^k(C^2).

Dominance order on Z^n.

(Nonstandard) Definition: a representation V of U(n) is pointed at lambda if lambda is a weight of V, its multiplicity is 1, and lambda dominates all the other weights of V.

Feb 1

Orthonormality of characters extends to measurable representations of groups for which one can define an invariant, finite, measure, e.g. S^1.
We know enough irreps of S^1, namely z |-> z^n for each integer n, that (by Fourier theory) we must have found all the characters of irreps. So that's them and they're all 1-dim.

Def. Algebraic subgroup of GL(n).
Def. Algebraic representation, meaning all the matrix coefficients are polynomials in the original matrix entries or det^{-1}.
(Changes of basis are given by linear polynomials, so the definition doesn't really depend on choice of basis.)

Non-ex. C acts on C by z |-> exp(lambda z). That's a smooth rep, but not algebraic.
Ex. SL(2) acting on Sym^2 C^2. The matrix entries are homogeneous quadratic.

Thm. Let N_+ be the upper triangular 2x2 matrices with 1s on the diagonal.
Then in any continuous representation, there are 1-dim invariant subspaces.
If the representation is algebraic, then those subspaces are trivial representations.

Let's look for representations of G = SL(2). They should all occur inside algebraic functions on SL(2), which is C[a,b,c,d] / (det = 1).
Rather, for each irrep V, we should find V @ V^*.
So take the 1 x N_+ invariants, to try to cut down V^* to something smaller, but by the theorem, nonzero.
Claim. The only 1 x N_+ invariant functions on SL(2) are C[a,c].
(We'll complete this proof next time.)

Cor. Each irrep occurs inside some Sym^k(C^2).
Next time: the Sym^k(C^2) are actually irreducible, and each has a unique N_+-invt vector.